                                               Homomorphism Proof Systems for Unsatisfiable Formulas

DOI：

 作者 单位 许道云 贵州大学,计算机科学系,贵州,贵阳,550025

合取范式(CNF)公式HF的同态φ是一个从H的文字集合到F的文字集合的映射,并保持补运算和子句映到子句.同态映射保持一个公式的不可满足性.一个公式是极小不可满足的是指该公式本身不可满足,而且从中删去任意一个子句后得到的公式可满足.MU(1)是子句数与变元数的差等于1的极小不可满足公式类.一个三元组(H,φ,F)称为的一个来自H的同态证明,如果φ是一个从H到F的同态.利用基础矩阵的方法证明了:一个不可满足公式F的树消解证明,可以在多项式时间内转换成一个来自MU(1)中公式的同态证明.从而,由MU(1)中的公式构成的同态证明系统是完备的,并且由MU(1)中的公式构成的同态证明系统与树消解证明系统之间是多项式等价的.

A homomorphism φ of CNF formulas from H to F is a function mapping the set of literals in H to the set of literals in F and it preserves complements and clauses. If the formula H is homomorphic to the formula F, then the unsatisfiability of H implies the unsatisfiability of F. A CNF formula F is minimally unsatisfiable if F is unsatisfiable and the resulting formula deleting any one clause from F is satisfiable. MU(1) is a class of minimally unsatisfiable formulas with the deficiency of the number of clauses and variables to be one. A triple (H,φ,F) is called a homomorphism proof from H of F if φ is a homomorphism from H to F. In this paper, a method from the basic matrix of MU(1) formula is used to prove that a tree resolution proof for an unsatisfiable formula F can be transformed into a homomorphism proof from a MU(1) formula for F. Whence, the homomorphism proof system from formulas in MU(1) is complete, and this proof system and the tree resolution proof system can be transformed mutually in polynomial time on the size of proof.
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